Hall's Random Walk is based on a questionable assumption and Flavin's imprecision on the mathematical summation of future incomes

Hall (1978) wrote a very short proof, i.e., a first order condition in the appendix (for a more detailed examination of consumption optimization using Euler equation I suggest reading the excellent Sargent's Macroeconomic Theory book). Writing out optimal consumption for period t and t+1, Flavin (1981) arrived at Hall's Random Walk result. However, it is possible to show that Flavin, for period t+1, had an imprecise index j for the summation of future incomes, without such error Hall's conclusion cannot hold. Further, under Clower's Dual-Decision Hypothesis section, we will show why Hall's Permanent Income Hypothesis assumption is questionable and impractical. Here is Flavin's approach and how she arrived at Hall's result:

Why Flavin's summation index at period t+1 is leads to a summation error!